Multi-view 3D displays are preferable to other stereoscopic display technologies because they provide autostereoscopic viewing from any viewpoint without special glasses. However, they require a large number of pixels to achieve high image quality. Therefore, data compression is a major issue for this approach. In this paper, we present a framework for efficient compression of multi-view video streams for multi-view 3D displays. Our goal is to optimize image quality without increasing the required data bandwidth. We achieve this by taking into account a precise notion of the multi-dimensional display bandwidth. The display bandwidth implies that scene elements that appear at a given distance from the display become increasingly blurry as the distance grows. Our main contribution is to enhance conventional multi-view compression pipelines with an additional pre-filtering step that bandlimits the multi-view signal to the display bandwidth. This imposes a shallow depth of field on the input images, thereby removing high frequency content. We show that this pre-filtering step leads to increased image quality compared to state-of-the-art multi-view coding at equal bitrate. We present results of an extensive user study that corroborate the benefits of our approach. Our work suggests that display pre-filtering will be a fundamental component in signal processing for 3D displays, and that any multi-view compression scheme will benefit from our pre-filtering technique.